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Let G / P be a rational homogeneous space (not necessarily irreducible) and $$x_0\in G/P$$ x 0 ∈ G / P be the point at which the isotropy group is P . The G -translates of the orbit $$Qx_0$$ Q x 0 of a parabolic subgroup $$Q\subsetneq G$$ Q ⊊ G such that $$P\cap Q$$ P ∩ Q is parabolic are called Q - cycles . We established an extension theorem for local biholomorphisms on G / P that map local pieces of Q -cycles into Q -cycles. We showed that such maps extend to global biholomorphisms of G / P if G / P is Q -cycle-connected, or equivalently, if there does not exist a non-trivial parabolic subgroup containing P and Q . Then we applied this to the study of local biholomorphisms preserving the real group orbits on G / P and showed that such a map extend to a global biholomorphism if the real group orbit admits a non-trivial holomorphic cover by the Q -cycles. The non-closed boundary orbits of a bounded symmetric domain embedded in its compact dual are examples of such real group orbits. Finally, using the results of Mok–Zhang on Schubert rigidity, we also established a Cartan–Fubini type extension theorem pertaining to Q -cycles, saying that if a local biholomorphism preserves the variety of tangent spaces of Q -cycles, then it extends to a global biholomorphism when the Q -cycles are positive dimensional and G / P is of Picard number 1. This generalizes a well-known theorem of Hwang–Mok on minimal rational curves.

Local holomorphic mappings respecting homogeneous subspaces on rational homogeneous spaces